Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{cos^2x-1}{cos\,x-1}\)

Question

Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x \cdot \sqrt{1 + x} \cdot (1 - x)^{3/2}}{dx} \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to:

Answer 1

Given:

lim_x→0 (cos²x − 1) / (cos x − 1)

Step 1: Factor the numerator

cos²x − 1 = (cos x − 1)(cos x + 1)

Step 2: Simplify the expression

= lim_x→0 (cos x − 1)(cos x + 1) / (cos x − 1)

Cancel (cos x − 1):

= lim_x→0 (cos x + 1)

Step 3: Apply the limit

As x → 0,
cos x → 1

Therefore,

lim_x→0 (cos x + 1) = 2

Final Answer:

lim_x→0 (cos²x − 1)/(cos x − 1) = 2

Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{cos^2x-1}{cos\,x-1}\)

Solution and Explanation

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